Harmonic analysis and arithmetic combinatorics

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Arithmetic combinatorics is an emerging area of mathematics which combines problems, ideas and methods from harmonic analysis, analytic and combinatorial number theory, combinatorics, and ergodic theory. It is organized around a number of central questions, such as the existence of patterns (e.g. arithmetic progressions) in large sets, or determining the structure of ``extremal" configurations (e.g. sets with small sumsets). Its recent highlights include the work of Gowers on Szemer\'edi's theorem and the Green-Tao result on long arithmetic progressions in the primes. My proposed research concerns the interconnections between harmonic analysis and various aspects of arithmetic combinatorics, in particular harmonic-analytic methods in additive number theory, connections between harmonic analysis and incidence geometry, and arithmetic-combinatorial methods in analysis and geometric measure theory.